#7624071

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#61479	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,8)(9,10,14,20)(11,23,12,17)(13,22,16,19)(15,30,29,33)(18,25,37,27)(21,28,39,32)(24,34,26,38)(31,35,40,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,14)(13,26)(15,25)(16,24)(18,30)(19,38)(20,23)(21,35)(22,34)(27,29)(28,31)(32,40)(33,37)(36,39);(9,12,14,11)(10,18,20,37)(13,26,16,24)(15,31,29,40)(17,33,23,30)(19,36,22,35)(21,34,39,38)(25,32,27,28);(2,5)(6,8)(9,13,14,16)(10,19,20,22)(11,24,12,26)(15,28,29,32)(17,34,23,38)(18,36,37,35)(21,33,39,30)(25,40,27,31);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,19,20,22)(11,24,12,26)(15,28,29,32)(17,34,23,38)(18,36,37,35)(21,33,39,30)(25,40,27,31)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531267);